# 18. SOLUTIONS TO EXERCISES

Exercise 2.1 Mean value and variance

Worksheet 2.1

 j L(j) - L(j) + dL F(j) 1 23.0-23.5 1 23.25 23.25 -2.968 8.809 2 23.5-24.0 1 23.75 23.75 -2.468 6.091 3 24.0-24.5 1 24.25 24.25 -1.968 3.873 4 24.5-25.0 2 24.75 49.50 -1.468 4.310 5 25.0-25.5 2 25.25 50.50 -0.968 1.874 6 25.5-26.0 6 25.75 154.50 -0.468 1.314 7 26.0-26.5 5 26.25 131.25 0.032 0.005 8 26.5-27.0 6 26.75 160.50 0.532 1.698 9 27.0-27.5 2 27.25 54.50 1.032 2.130 10 27.5-28.0 2 27.75 55.50 1.532 4.694 11 28.5-29.0 2 28.25 56.50 2.032 8.258 12 28.5-29.0 1 28.75 28.75 2.532 6.411 sums 31 812.75 49.467 s2 = 1.6489 s = 1.2841

Exercise 2.2 The normal distribution

Worksheet 2.2

 x Fc(x) x Fc(x) 22.0 0.02 26.0 4.75 22.5 0.07 26.5 4.70 23.0 0.21 27.0 4.00 23.5 0.51 27.5 2.93 24.0 1.08 28.0 1.84 24.5 1.97 28.5 0.99 25.0 3.07 29.0 0.46 25.5 4.12 29.5 0.18

Fig. 18.2.2 Bell-shaped curve determined for length-frequency sample of Fig. 17.2.1

Exercise 2.3 Confidence limits

L - t30 * s/Ö n = 26.22 - 2.04 * 1. 284/Ö 31 = 25.75

L + t30 * s/Ö n = 26.22 - 2.04 * 1. 284/Ö 31 = 26.69

Exercise 2.4 Ordinary linear regression analysis

Worksheet 2.4

 year i number of boats catch per boat per year x(i) x(i)2 y(i) y(i)2 x(i) * y(i) 1971 1 456 207936 43.5 1892.25 19836.0 1972 2 536 287296 44.6 1989.16 23905.6 1973 3 554 306916 38.4 1474.56 21273.6 1974 4 675 455625 23.8 566.44 16065.0 1975 5 702 492804 25.2 635.04 17690.4 1976 6 730 532900 30.5 930.25 22265.0 1977 7 750 562500 27.4 750.76 20550.0 1978 8 918 842724 21.1 445.21 19369.8 1979 9 928 861184 26.1 681.21 24220.8 1980 10 897 804609 28.9 835.21 25923.3 Total 7146 5354494 309.5 10200.09 211099.5 sx = 165.99 sy = 8.307 variance of b: sb = 0.01034 variance of a: sa = 7.568 Student's distribution: tn-2 = 2.31 confidence limits: b - sb * tn-2, b + sb * tn-2 = [-0.0645, -0-0167] a - sa * tn-2, a + sa * tn-2 = [42.5,77.4]

Exercise 2.5 The correlation coefficient

In principle the number of boats can be measured with any accuracy, so this is the natural independent variable. The correlation coefficient is not considered useful in the present context. Nevertheless, as an exercise we calculate the confidence limits, using Eqs. 2.5.3 in sections called A and B:

A = 0.5 * ln[(1 + r)/(1 - r)] = 0.5 * ln[(1 - 0.811)/(1 + 0.811)] = -1.130

r1 = tanh(A - B) = -0.95

r2 = tanh(A + B) = -0.37

Exercise 2.6a Linear transformations, the Bhattacharya plot Worksheet 2.6a

 x F(x) ln F(x) D ln F(z) x + dL/2 remarks (y) (z) 4.5 2 0.693 not used 0.916 5 5.5 5 1.609 0.875 6 6.5 12 2.485 0.693 7 7.5 24 3.178 0.377 8 8.5 35 3.555 0.182 9 9.5 42 3.737 not used contaminated 0.000 10 10.5 42 3.737 0.091 11 11.5 46 3.829 0.197 12 12.5 56 4.025 0.035 13 13.5 58 4.060 not used -0.254 14 14.5 45 3.807 -0.716 15 15.5 22 3.091 -1.145 16 16.5 7 1.946 -1.253 17 17.5 2 0.693

 First component Second component intercept (a) 2.328 5.978 slope (b) -0.240 -0.446 9.7 13.4 s2 = - 1/b 4.18 2.24 s 2.04 1.50

Worksheet 2.6b

 First component Second component B = -1/(2 * 2.042) = -0.120 B = -1/(2 * 1.502) = -0.222

 x Fc(x)first Fc(x)second x Fc(x)first Fc(x)second 1.5 0.0 11.5 26.4 23.7 2.5 0.1 12.5 15.2 44.2 3.5 0.4 13.5 6.9 52.8 4.5 1.5 14.5 2.4 40.4 5.5 4.7 15.5 0.7 19.9 6.5 11.4 16.5 0.2 6.3 7.5 21.8 0.0 17.5 0.0 1.3 8.5 32.7 0.3 18.5 0.2 9.5 38.7 1.8 19.5 0.0 10.5 36.0 8.2 20.5

Fig. 18.2.6A Bhattacharya plots (linear transformations) (see Worksheet 2.6a)

Exercise 3.1 The von Bertalanffy growth equation Worksheet 3.1

 ageyears standard lengthcm total lengthcm body weightg 0.5 1.0 1.4 0.04 1.0 6.6 8.0 9 1.5 11.8 14.1 45 2 16.5 19.7 118 3 24.9 29.6 380 4 32.0 37.9 775 5 38.0 45.0 1262 6 43.0 51.0 1802 7 47.3 56.0 2359 8 50.9 60.3 2909 9 54.0 63.9 3434 10 56.6 67.0 3922 12 60.6 71.7 4770 14 63.5 75.1 5444 16 65.5 77.5 5961 20 68.1 80.5 6637 50 70.7 83.6 7388

Fig. 18.3.1 Growth curves based on von Bertalanffy growth equations

Exercise 3.1.2 The weight-based von Bertalanffy growth equation Worksheet 3.1.2

 aget lengthL (t) weightw (t) aget lengthL (t) weightw (t) 0 2.54 0.38 0.9 9.34 19.00 0.1 3.63 1.11 1.0 9.78 21.83 0.2 4.62 2.29 1.2 10.55 27.36 0.3 5.51 3.90 1.4 11.17 32.53 0.4 6.32 5.88 1.6 11.69 37.21 0.5 7.05 8.16 1.8 12.11 41.37 0.6 7.71 10.69 2.0 12.45 44.99 0.7 8.31 13.37 2.5 13.06 51.93 0.8 8.85 16.16 3.0 13.43 56.47

Fig. 18.3.1.2 Growth curves for ponyfish

Exercise 3.2.1 Data from age readings and length compositions (age/length key) Worksheet 3.2.1

 cohort 1982S 1981A 1981S 1980A number in length sample 1982S 1981A 1981S 1980A length interval key numbers per cohort 35-36 0.800 0.200 0 0 53 42.4 10.6 0 0 36-37 0.636 0.273 0.091 0 61 38.8 16.7 5.6 0 37-38 0.600 0.300 0.100 0 49 29.4 14.7 4.9 0 38-39 0.500 0.400 0.100 0 52 26.0 20.8 5.2 0 39-40 0.364 0.364 0.182 0.091 70 25.5 25.5 12.7 6.4 40-41 0.273 0.455 0.182 0.091 52 14.2 23.7 9.5 4.7 41-42 0.222 0.444 0.222 0.111 49 10.9 21.8 10.0 5.4 total 386 187.2 133.8 48.8 16.5

Exercise 3.3.1 The Gulland and Holt plot

Worksheet 3.3.1

 A B C D E F fishno. L(t)cm L(t + D t)cm D tdays cm/year(y) cm(x) 1 9.7 10.2 53 3.44 9.95 2 10.5 10.9 33 4.42 10.70 3 10.9 11.8 108 3.04 11.35 4 11.1 12.0 102 3.22 11.55 5 12.4 15.5 272 4.16 13.95 6 12.8 13.6 48 6.08 13.20 7 14.0 14.3 53 2.07 14.15 8 16.1 16.4 73 1.50 16.25 9 16.3 16.5 63 1.16 16.40 10 17.0 17.2 106 0.69 17.10 11 17.7 18.0 111 0.99 17.85 a (intercept) = 8.77 b (slope) = -0.431 K = -b = 0.43 per year L¥ = -a/b = 20.3 cm sb = 0.145 t9 = 2.26 confidence interval of K = [0.10, 0.76]

Fig. 18.3.3.1 Gulland and Holt plot (see Worksheet 3.3.1)

Exercise 3.3.2 The Ford-Walford plot and Chapman's method

Worksheet 3.3.2

 Plot FORD-WALFORD CHAPMAN t L(t)(x) L(t + D t)(y) L(t)(x) L(t + D t) - L(t)(y) 1 35 55 35 20 2 55 75 55 20 3 75 90 75 15 4 90 105 90 15 5 105 115 105 10 a (intercept) 26.2 26.2 b (slope) 0.86 -0.14 0.0009268 0.0009271 0.030 0.030 tn-2 3.18 3.18 confidence limits of b [0.76, 0.96] [-0.24, -0.04] K - ln b/D t = 0.15 -(1/1) * ln (1 + b) = 0.15 L¥ 1/(1 - b) = 185 cm -a/b = 185 cm

Fig. 18.3.3.2 Ford-Walford and Chapman plots for yellowfin tuna off Senegal. Data source: Postel, 1955, (see Worksheet 3.3.2)

Ford-Walford plot

Chapman's method

Exercise 3.3.3 The von Bertalanffy plot

We choose 11 inches as estimate for L¥ , because very few (1.5%) of the seabreams are longer than 11 inches.

We assign the arbitrary ages of 1,2,3 and 4 years to the four age groups.

 age L -ln (1 - L/L¥ ) 1 3.22 0.35 2 5.33 0.66 3 7.62 1.18 4 9.74 2.17

b (slope) = K = 0.60 per year

At least, K has now got the correct sign.

sb2 = 0.0119, sb = 0.109, t2 = 4.3

confidence interval of K= [0.13, 1.07]

t0 cannot be estimated because the absolute age is not known.

Fig. 18.3.3.3 Von Bertalanffy and Gulland and Holt plots for sea breams. Data source: Cassie, 1954

von Bertalanffy plot

Gulland and Holt plot

Exercise 3.4.1 Bhattacharya's method

There is no "correct" solution to this exercise. The following is a "suggestion for a solution". It is not the same result as the one obtained by Weber and Jothy (1977) by using the Cassie method.

Worksheet 3.4.1a

 A B C D E F G H I length interval N1+ ln N1+ D ln N1+(y) L(x) D ln N1 ln N1 N1 N2+ 5.75-6.75 1 0 - - - - 1 0 6.75-7.75 26 3.258 (3.258) 6.75 1.262 - 26 0 7.75-8.75 42# 3.738# 0.480 7.75 0.354 3.738# 42# 0 8.75-9.75 19 2.944 -0.793 8.75 -0.554 3.183 19 0 9.75-10.75 5 1.609 -1.335* 9.75 -1.462 1.722 5 0 10.75-11.75 15 2.708 1.099 10.75 - -0.648 0.5 14.5 11.75-12.75 41 3.714 1.006 11.75 2.370 -3.926 0.0 41.0 12.75-13.75 125 4.828 1.115 12.75 -3.278 - - 125 13.75-14.75 135 4.905 0.077 13.75 - - - 135 .......... .......... .......... - Total 1069 93.5 a (intercept) = 7.391 b (slope) = -0.908 *) points used in the regression analysis# clean starting point

Worksheet 3.4.1b

 A B C D E F G H I interval N2+ ln N2+ D ln N2+ L D ln N2 ln N2 N2 N3+ ...... ..... 10.75-11.75 14.5 2.674 - 10.75 - - 14.5 0 11.75-12.75 41 3.714 1.039* 11.75 - - 41 0 12.75-13.75 125# 4.828# 1.115* 12.75 - 4.828# 125# 0 13.75-14.75 135 4.905 0.077* 13.75 0.238 5.066 135 0 14.75-15.75 102 4.625 -0.280* 14.75 -0.262 4.806 102 0 15.75-16.75 131 4.875 0.250 15.75 -0.761 4.843 57.0 74.0 16.75-17.75 106 4.663 -0.212 16.75 -1.261 4.043 16.2 89.8 17.75-18.75 86 4.454 -0.209 17.75 -1.760 2.782 2.8 83.2 18.75-19.75 59 4.078 -0.377 18.75 -2.260 1.022 0.3 58.7 19.75-20.75 43 3.761 -0.316 19.75 -2.759 -1.038 0.0 43 20.75-21.75 45 3.807 0.045 20.75 - -3.997 - 45 21.75-22.75 56 4.025 0,219 21.75 - - - 56 ...... ..... Total 493.8 a (intercept) = 7.11 b (slope) = -0.500

Worksheet 3.4. 1c

 A B C D E F G H I interval N3+ ln N3+ D ln N3+ L D ln N3 ln N3 N3 N4+ ...... ..... 15.75-16.75 74.0 - - 15.75 - - 74 0 16.75-17.75 89.8 4.498 0.194* 16.75 - - 89.9 0 17.75-18.75 83.2# 4.421# -0.076* 17.75 - 4.421# 83.2# 0 18.75-19.75 58.7 4.072 -0.348* 18.75 -0.225 4.196 58.7 0 19.75-20.75 43 3.761 -0.312* 19.75 -0.404 3.792 43.0 0 20.75-21.75 45 3.807 0.046 20.75 -0.583 3.209 24.8 20.2 21.75-22.75 56 4.025 0.219 21.75 -0.762 2.447 11.6 44.4 22.75-23.75 20 2.996 -1.030 22.75 -0.941 1.506 4.5 15.5 23.75-24.75 8 2.079 -0.916 23.75 -1.120 0.386 1.5 6.5 24.75-25.75 3 1.099 -0.981 24.75 -1.299 -0.913 0.4 2.6 25.75-26.75 1 0 -1.099 25.75 - - - 1 Total 391.5 a (intercept) = 3.13 b (slope) = -0.179

Worksheet 3.4.1d

 A B C D E F G H I interval N2+ ln N2+ D ln N2+ L D ln N2 ln N2 N2 N3+ ...... ..... 20.75-21.75 20.2 3.006 - 20.75 ? too few observations 21.75-22.75 44.4 3.793 0.787 21.75 ? 22.75-23.75 15.5 2.741 -1.052 22.75 ? 23.75-24.75 6.5 1.892 -0.869 23.75 ? 24.75-25.75 2.6 0.956 -0.916 24.75 ? 25.75-26.75 1 0 -0.956 25.75 ?

Gulland and Holt plot:

 age D L/D t L 1 8.1 6.1 11.15 2 14.2 3.3 15.85 3 17.5

a (intercept) = 12.7
K = -b = 0.60 per year
b (slope) = -0.60
L¥ = -a/b = 21.4 cm

Fig. 18.3.4.1B Gulland and Holt plot of mean lengths of cohorts obtained by the Bhattacharya method (see Worksheets 3.4.1a, b, and c and Fig. 18.3.4.1A)

Exercise 3.4.2 Modal progression analysis

A. Leiognathus splendens:

Worksheet 3.4.2

 GULLAND AND HOLT PLOT VON BERTALANFFY PLOT time of sampling L (t) D L/D t L t -ln (1 - L/L¥ ) 1 June 2.8 0.42 0.325 6.8 3.65 1 Sep. 4.5 0.67 0.590 5.2 5.15 1 Dec. 5.8 0.92 0.854 4.0 6.30 1 March 6.8 1.17 1.119 a (intercept) 10.65 -0.12 b (slope, -K or K) -1.06 1.06 -a/b L¥ = 10.1 t0 = 0.11 L (t) = 10.1 * [1 - exp(-1.1 * (t - 0.11))]

Fig. 18.3.4.2A Modal progression in time series of length-frequencies of ponyfish. (See Worksheet 3.4.2)

B. Rastrelliger kanagurta:

 GULLAND AND HOLT PLOT VON BERTALANFFY PLOT time of sampling L (t) D L/D t L t -ln (1 - L/L¥ ) 1 Feb 13.3 0.08 0.648 21.6 14.20 1 March 15.1 0.17 0.779 17.4 16.55 1 May 18.0 0.33 1.036 16.8 18.70 1 June 19.4 0.42 1.189 13.2 19.95 1 July 20.5 0.50 1.327 9.6 20.9 1 August 21.3 0.58 1.442 a (intercept) 44.57 0.512 b (slope, -K or K) -1.60 1.61 -a/b L¥ = 27.9 t0 = -0.32 L(t) = 27.9 * [1 - exp(-1.6 * (t + 0.32))]

Fig. 18.3.4.2B Modal progression in time series of length-frequencies of Indian mackerel. (See Worksheet 3.4.2)

Exercise 3.5.1 ELEFAN I

Worksheet 3.5.1

 RESTRUCTURING OF LENGTH FREQUENCY SAMPLE STEP1 STEP2 STEP3 STEP4a STEP4b STEP5 STEP6 mid-lengthL orig. freq.FRQ(L) MA(L) FRQ/MA zeroes deemphasized points highest positive points 5 4 4.6 0.870 -0.197 2 -0.197 -0.109 10 13 4.6 2.826 1.610 2 0.966 0.966 0.966 15 6 4.8 1.250 0.154 1 0.123 0.123 20 0 4.0 0 -1.000 1 -1.000 0 25 1 1.4 0.714 -0.341 3 -0.340 -0.188 30 0 0.4 0 -1.000 2 -1.000 0 35 0 1.0 0 -1.000 1 -1.000 0 40 1 1.0 1.000 -0.077 2 -0.077 -0.043 45 3 1.0 3.000 1.770 2 1.062 1.062 1.062 50 1 1.2 0.833 -0.231 1 -0.230 -0.127 55 0 1.0 0 -1.000 1 -1.000 0 60 1 0.4 2.500 1.308 3 0.523 0.523 0.523 S = 12.993 SP = 2.674 (S /12) = M = 1.083 SN = -4.845 ASP = 2.551 -SP/SN = R = 0.552

Exercise 3.5.1a ELEFAN I, continued

Worksheet 3.5.1a

 RESTRUCTURING OF LENGTH FREQUENCY SAMPLE STEP1 STEP2 STEP3 STEP4a STEP4b STEP5 STEP6 mid-lengthL orig. freq.FRQ(L) MA(L) FRQ/MA zeroes deemphasized points highest positive points 20 14 18.2 0.769 -0.194 2 -0.194 -0.159 24 32 40.0 0.800 -0.162 1 -0.162 -0.133 28 45 63.0 0.714 -0.252 0 -0.252 -0.206 32 109 75.8 1.438 0.506 0 0.506 0.506 36 115 78.4 1.467 0.537 0 0.537 0.537 0.537 40 78 75.2 1.037 0.086 0 0.086 0.086 44 45 58.0 0.776 -0.187 0 -0.187 -0.153 48 29 37.2 0.780 -0.183 0 -0.183 -0.150 52 23 24.0 0.958 0.003 0 0.003 0.003 0.003 56 11 16.0 0.688 -0.279 0 -0.279 -0.228 60 12 10.6 1.132 0.186 0 0.186 0.186 0.186 64 5 6.2 0.806 -0.156 0 -0.156 -0.128 68 2 4.4 0.455 -0.523 0 -0.523 -0.428 72 1 2.0 0.500 -0.476 1 -0.476 -0.390 76 2 1.0 2.000 1.095 2 0.657 0.657 0.657 S = 14.320 SP = 1.975 (S /15) = M = 0.9547 SN = -2.413 ASP = 1.383 -SP/SN = R = 0.818

Fig 18.3.5.1A Regrouped length-frequency data of 523 pike (4 cm length intervals), ELEFAN I restructured data and highest positive points and mean lengths as determined from age reading (low arrow). (See Worksheet 3.5.1a, cf. Fig 17.3.5.1C)

Exercise 4.2 The dynamics of a cohort (exponential decay model with variable Z)

Worksheet 4.2

 age groupt1 - t2 M F Z e-0.5z N(t1) N(t2) N(t1) - N(t2) F/Z C(t1, t2) 0.0-0.5 2.0 0.0 2.0 0.3679 10000 3679 6321 0 0 0.5-1.0 1.5 0.0 1.5 0.4724 3679 1738 1941 0 0 1.0-1.5 0.5 0.2 0.7 0.7047 1738 1225 513 0.286 147 1.5-2.0 0.3 0.4 0.7 0.7047 1225 863 362 0.571 207 2.0-2.5 0.3 0.6 0.9 0.6376 863 550 313 0.667 209 2.5-3.0 0.3 0.6 0.9 0.6376 550 351 199 0.667 133 3.0-3.5 0.3 0.6 0.9 0.6376 351 224 127 0.667 85 3.5-4.0 0.3 0.6 0.9 0.6376 224 143 81 0.667 54 4.0-4.5 0.3 0.6 0.9 0.6376 143 91 52 0.667 35 4.5-5.0 0.3 0.6 0.9 0.6376 91 58 33 0.667 22

Exercise 4.2a The dynamics of a cohort (the formula for average number of survivors, Eq. 4.2.9)

The formula for average number of survivors (Eq. 4.2.9).

Exact value:

Approximation:

Exercise 4.3 Estimation of Z from CPUE data

Worksheet 4.3

 cohort 1982 A 1982 S 1981 A 1981 S 1980 A age t2 1.14 1.64 2.14 2.64 3.14 CPUE 111 67 40 24 15 cohort age t1 CPUE 1983 S 0.64 182 0.99 1.00 1.01 1.01 1.00 1982 A 1.14 111 ------ 1.03 1.02 1.02 1.00 1982 S 1.64 67 ------ ------ 1.03 1.03 1.00 1981 A 2.14 40 ------ ------ ------ 1.02 0.98 1981 S 2.64 24 ------ ------ ------ ------ 0.94

Exercise 4.4.3 The linearized catch curve based on age composition data

Worksheet 4.4.3

 aget(x) yeary C(y, t, t + 1) ln C(y, t, t + 1)(y) remarks 0 1974 599 6.395 not used 1 1975 860 6.757 2 1976 1071 6.976 3 1977 269 5.596 used in the analysis 4 1978 69 4.234 5 1979 25 3.219 6 1980 8 2.079 7 1981 - - slope: b = -1.16 sb2 = [(sy/sx)2 - b2]/(n - 2) = 0.002330 sb = 0.0483 sb * tn-2 = 0.0483 * 4.30 = 0.21 Z = 1.16 ± 0.21

Fig. 18.4.4.3 The linearized catch curve based on age composition data (see Worksheet 4.4.3)

Exercise 4.4.5 The linearized catch curve based on length composition data Worksheet 4.4.5

 L1 - L2 C(L1, L2) t(L1) D t (x) (y) z(slope) remarks 7-8 11 0.452 0.0759 0.489 4.976 - not used 8-9 69 0.527 0.0796 0.567 6.765 - 9-10 187 0.607 0.0836 0.648 7.712 - 10-11 133 0.691 0.0881 0.734 7.319 - 11-12 114 0.779 0.0931 0.825 7.110 - 12-13 261 0.872 0.0987 0.921 7.880 - 13-14 386 0.971 0.1050 1.022 8.210 - 14-15 445 1.076 0.112 1.13 8.286 - 15-16 535 1.188 0.120 1.25 8.400 - used in analysis 16-17 407 0.308 0.130 1.37 8.051 - 17-18 428 1.438 0.141 1.51 8.019 1.43 18-19 338 1.579 0.154 1.65 7.693 1.60 19-20 184 1.733 0.170 1.82 6.987 2.27 20-21 73 1.903 0.190 2.00 5.953 3.07 21-22 37 2.092 0.214 2.20 5.152 3.45 22-23 21 2.307 0.246 2.43 4.446 3.54 23-24 19 2.553 0.290 2.69 4.183 3.30 24-25 8 2.843 0.352 3.01 3.124 3.20 25-26 7 3.195 0.448 3.40 2.749 - too close to L¥ 26-27 2 3.643 0.617 3.92 1.176 -

Details of the regression analyses:

 length group slope number of observations Student's distrib. variance of slope stand. dev. of slope confidence limits of Z L1 - L2 Z n tn-2 sb2 sb Z ± tn-2 * sb 15-16 - 1 - - - - 16-17 - 2 - - - - 17-18 1.43 3 12.70 0.59 0.7681 1.43 ± 9.75 18-19 1.60 4 4.30 0.12 0.3464 1.60 ± 1.49 19-20 2.27 5 3.18 0.156 0.3950 2.27 ± 1.26 20-21 3.07 6 2.78 0.228 0.4475 3.07 ± 1.33 21-22 3.45 7 2.57 0.140 0.3742 3.45 ± 0.96 22-23 3.54 8 2.45 0.071 0.2665 3.54 ± 0.65 23-24 3.30 9 2.37 0.051 0.2258 3.30 ± 0.54 24-25 3.20 10 2.31 0.030 0.1732 3.20 ± 0.40

Fig. 18.4.4.5 The linearized catch curve based on length composition data (see Worksheet 4.4.5)

Fig. 18.4.4.6 The cumulated catch curve based on length composition data (Jones and van Zalinge method) (see Worksheet 4.4.6)

Exercise 4.4.6 The cumulated catch curve based on length composition data (the Jones and van Zalinge method)

Worksheet 4.4.6

 L1 - L2 C(L1, L2) S C (L1, L¥ ) cumulated ln S C (L1, L¥ )(y) ln (L¥ - L1)(x) Z/K(slope) remarks 7-8 11 3665 8.207 3.100 - not used, not under full exploitation 8-9 69 3654 8.204 3.054 - 9-10 187 3585 8.185 3.006 - 10-11 133 3398 8.131 2.955 - 11-12 114 3265 8.091 2.901 - 12-13 261 3151 8.055 2.845 - 13-14 386 2890 7.969 2.785 - 14-15 445 2504 7.825 2.721 - 15-16 535 2059 7.630 2.653 - used in analysis 16-17 407 1524 7.329 2.580 - 17-18 428 1117 7.018 2.501 4.03 18-19 338 689 6.565 2.416 4.56 19-20 184 351 5.861 2.322 5.28 20-21 73 167 5.118 2.219 5.81 21-22 37 94 4.543 2.104 5.86 22-23 21 57 4.043 1.974 5.62 23-24 19 36 3.584 1.825 5.25 24-25 8 17 2.833 1.649 5.00 25-26 7 9 2.197 1.435 - too close to L¥ 26-27 2 2 0.693 1.163 -

Details of the regression analyses:

 length group slope* K number of obs. Student's distrib. variance of slope stand. dev. of slope confidence limits of Z L1 - L2 Z n tn-2 sb2 sb Z ± K * tn-2 * sb 15-16 - 1 - - - - 16-17 - 2 - - - - 17-18 2.44 3 12.70 0.00289 0.05376 2.44 ± 0.41 18-19 2.77 4 4.30 0.858 0.2929 2.77 ± 0 76 19-20 3.20 5 3.18 0.169 0.4111 3.20 ± 0.79 20-21 3.52 6 2.78 0.141 0.3755 3.52 ± 0.63 21-22 3.55 7 2.57 0.064 0.2530 3.55 ± 0.39 22-23 3.41 8 2.45 0.045 0.2121 3.41 ± 0.32 23-24 3.20 9 2.37 0.056 0.2366 3.20 ± 0.34 24-25 3.03 10 2.31 0.045 0.2121 3.03 ± 0.30

Exercise 4.4.6a The Jones and van Zalinge method applied to shrimp

Worksheet 4.4.6a

 carapace length mm numbers landed/year(millions) cumulated numbers/year(millions) remarks L1 - L2 C (L1, L2) S C (L1, L¥ ) ln S C (L1, L¥ )(y) ln (L¥ - L1)(X) Z/K(slope) 11.18-18.55 2.81 18.16 2.899 3.592 - not used 18.55-22.15 1.30 15.35 2.731 3.366 - 22.15-25.27 2.96 14.05 2.643 3.233 - 25.27-27.58 3.18 11.09 2.406 3.101 - used in analysis 27.58-29.06 2.00 7.91 2.068 2.992 - 29.06-30.87 1.89 5.91 1.777 2.915 3.36 30.87-33.16 1.78 4.02 1.391 2.811 3.52 33.16-36.19 0.98 2.24 0.806 2.663 3.68 36.19-40.50 0.63 1.26 0.231 2.426 3.32 40.50-47.50 0.63 0.63 -0.462 1.946 too close to L¥

Details of the regression analysis:

 lower length slope number of obs. Student's distrib. variance of slope stand. dev. of slope confidence limits of slope L1 Z/K n tn-2 sb2 sb Z/K ± tn-2 * sb 29.06 3.36 3 12.70 0.0354 0.1882 3.36 ± 2.39 30.87 3.52 4 4.30 0.0143 0.1196 3.52 ± 0.51 33.16 3.68 5 3.18 0.0096 0.0980 3.68 ± 0.31 36.19 3.32 6 2.78 0.0224 0.1497 3.32 ± 0.42

Fig. 18.4.4.6A Cumulated catch curve based on industrial shrimp fisheries in Kuwait. Data source: Jones and van Zalinge, 1981 (see Worksheet 4.4.6a)

Exercise 4.5.1 Beverton and Holt's Z-equation based on length data (applied to shrimp)

Worksheet 4.5.1

 A B C D E F G H carapace length groupmm numbers landed/year(millions) cumulated catch mid-length *) *) *) *) remarks L' (L1) - L2 C (L1, L2) S C (L1, L¥ ) Z/K 11.18-18.55 2.81 18.16 14.87 41.77 478.56 26.35 1.39 not used 18.55-22.15 1.30 15.35 20.35 26.46 436.79 28.46 1.92 22.15-25.27 2.96 14.05 23.71 70.18 410.33 29.21 2.59 25.27-27.58 3.18 11.09 26.43 84.03 340.15 30.67 3.12 27.58-29.06 2.00 7.91 28.32 56.64 256.12 32.38 3.15 29.06-30.87 1.89 5.91 29.97 56.63 199.48 33.75 2.93 30.87-33.16 1.78 4.02 32.02 56.99 142.85 35.53 2.57 33.16-36.19 0.98 2.24 34.68 33.98 85.86 38.33 1.77 36.19-40.50 0.63 1.26 38.35 24.16 51.88 41.17 1.27 numbers too low 40.50-47.50 0.63 0.63 44.00 27.72 27.72 44.00 1.00

Exercise 4.5.4 The Powell-Wetherall method

Worksheet 4.5.4

 A B C D *) E *) F *) G *) H *) L' (L1) - L2 C (L1, L2) (% catch) S C(L',¥)(% cumulated) (x) (y) 14-15 1.8 14.5 100.1 26.10 2086.95 20.849 6.849 15-16 3.4 15.5 98.3 52.70 2060.85 20.965 5.965 16-17 5.8 16.5 94.9 95.70 2008.15 21.161 5.161 17-18 8.4 17.5 89.1 147.00 1912.45 21.646 4.464 18-19 9.1 18.5 80.7 168.35 1765.45 21.877 3.877 19-20 10.2 19.5 71.6 198.90 1597.10 22.306 3.306 20-21 *) 14.3 20.5 61.4 293.15 1398.20 22.772 2.772 21-22 *) 13.7 21.5 47.1 294.55 1105.10 23.463 2.463 22-23 *) 10.0 22.5 33.4 225.00 810.50 24.266 2.266 23-24 *) 6.3 23.5 23.4 148.05 585.50 25.021 2.021 24-25 *) 6.4 24.5 17.1 156.80 437.45 25.582 1.582 25-26 *) 5.3 25.5 10.7 135.15 280.65 26.229 1.229 26-27 *) 3.3 26.5 5.4 87.45 145.5 26.944 0.944 27-28 *) 1.8 27.5 2.1 49.50 58.05 27.643 0.643 28-29 *) 0.3 28.5 0.3 8.55 8.55 28.500 0.500 b (slope) = -0.2997 a (intercept) = 8.795 Z/K = -(1 +b)/b = 2.337 L¥ = -a/b = 29.35 *) Considered fully recruited (n = 9)Steady state with constant parameter system.

Comment:

Back in 1974, when Munro (1983) reported on the grunts, it was not easy to estimate L¥ (ELEFAN etc. was not available). The Ford-Walford plot resulted in almost parallel lines for all species and, consequently, could not produce reliable estimates of their L¥ . Based on modal progression analysis, Munro instead, obtained by trial-and-error, the value of L¥ which seemed to produce a straight line in the von Bertalanffy plot. The result was L¥ = 40 cm producing K = 0.26 per year. Using L' = 20 cm he then obtained Z/K = (40 - 22.772)/2.772 = 6.2 from Beverton and Holt's formula. (This estimate represents the straight line on the plot that connects the L' = 20 cm point with an x-intercept of L¥ = 40 cm, i.e. a line with slope b = -(1 + Z/K)-1 = -0.14.) Thus, Munro obtained Z = 6.2 * 0.26 = 1.6 per year. However, a L¥ » 30 cm changes Munro's MPA somewhat and using his procedure one cannot reject L¥ » 30 cm and K » 0.5 per year. Using our results we then obtain Z = 2.34 * 0.5 = 1.17 per year.

Exercise 4.6 Plot of Z on effort (estimation of M and q)

Worksheet 4.6

 year effort mean length *) cm 1966 2.08 15.7 1.97 1967 2.80 15.5 2.05 1968 3.50 16.1 1.82 1969 3.60 14.9 2.32 1970 3.80 14.4 2.58 1071 no data 1972 no data 1973 9.94 12.8 3.74 1974 6.06 12.8 3.74 *) in millions of trawling hours

L¥ = 29.0 cm
K = 1.2 per year
Lc = 7.6 cm

a) Based on data for the years 1966-1970:

slope: q = 0.23 ± 0.66
sq2 = 0.0424
sq = 0.206
t3 * sq = 3.18 * 0.206 = 0.66

intercept: M = 1.41 ± 2.11
sM2 = 0.439
sM = 0.663
t3 * sM = 3.18 * 0.663 = 2.11

Both confidence intervals contain 0 and negative values which makes no biological sense. The variation in effort is too small to support a dependable regression analysis.

b) Based on data for the years 1966-1974:

slope: q = 0.27 ± 0.17
sq2 = 0.00429
sq = 0.0655
t5 * sq = 2.57 * 0.0655 = 0.17

intercept: M = 1.39 ± 0.87
sM2 = 0.115
sM = 0.339
t5 * sM = 2.57 * 0.339 = 0.87

Fig. 18.4.6 Plot of Z on effort, to estimate M and q of Priacanthus sp. Data source: Boonyubol and Hongskul, 1978 (see Worksheet 4.6)

Exercise 5.2 Age-based cohort analysis (Pope's cohort analysis)

a) terminal

F = F6 = 1.0
C6 = 8

 C5 = 25 N5 = 44.4 F5 = 0.97 C4 = 69 N4 = 130.4 F4 = 0.88 C3 = 269 N3 = 456.6 F3 = 1.05 C2 = 1071 N2 = 1741.3 F2 = 1.14 C1 = 860 N1 = 3077.3 F1 = 0.37 C0 = 599 N0 = 4420.7 F0 = 0.16

b) terminal

F = F6 = 2.0
C6 = 8

 C5 = 25 N5 = 39.7 F5 = 1.18 C4 = 69 N4 = 124.8 F4 = 0.94 C3 = 269 N3 = 449.7 F3 = 1.08 C2 = 1071 N2 = 1732.9 F2 = 1.15 C1 = 860 N1 = 3067.3 F1 = 0.37 C0 = 599 N0 = 4408.0 F0 = 0.16

Fig. 18.5.2 Pope's (age-based) cohort analysis of whiting, with different values of terminal F, to demonstrate VPA convergence. Data source: ICES, 1981

Exercise 5.3 Jones' length-based cohort analysis

Worksheet 5.3

 length group natural mortality factor number caught (mill.) number of survivors exploitation rate fishing mortality total mortality L1 - L2 H(L1, L2) C(L1, L2) N(L1) F/Z F Z 11.18-18.55 1.1854 2.81 119.82 0.08 0.32 4.22 18.55-22.15 1.1047 1.30 82.90 0.08 0.34 4.24 22.15-25.27 1.1035 2.96 66.75 0.20 0.99 4.89 25.27-27.58 1.0858 3.18 52.13 0.29 1.62 5.52 27.58-29.06 1.0596 2.00 41.29 0.31 1.77 5.67 29.06-30.87 1.0806 1.89 34.89 0.28 1.51 5.41 30.87-33.16 1.1175 1.78 28.13 0.25 1.28 5.18 33.16-36.19 1.1949 0.98 20.93 0.14 0.63 4.53 36.19-40.50 1.4331 0.63 13.84 0.08 0.36 4.26 40.50-47.50 - 0.63 6.30 0.10 0.43 *) 4.33 *) F (40.50 - 47.50) = 3.9 * 0.1/(1 - 0.1) = 0.43

The cumulated catch curve (Exercise 4.4.6a) gave a Z/K value of about 3.

From this we have Z = 3 * 2.6 = 7.8; F = Z-M = 7.8-3.9 = 3.9; exploitation rate, F/Z = 3.9/7.8 = 0.5

Exercise 6.1 A mathematical model for the selection ogive

L50% = 13.6 cm
S1 = 13.6 * ln (3)/(14.6 - 13.6) = 14.941

L75% = 14.6 cm
S2 = ln (3)/(14.6 - 13.6) = 1.0986

S (L) = 1/[1 + exp(14.941 - 1.0986 * L)]

 L 11 12 13 14 15 16 17 18 S(L) 0.05 0.15 0.34 0.61 0.82 0.93 0.98 0.99

Fig. 18.6.1 Length-based selection ogive

Exercise 6.5 Estimation of the selection ogive from a catch curve

Worksheet 6.5

 A B C D E F G H I length groupL1 - L2 ta) D t C(L1, L2) ln (C/D t)b) St obs.c) ln (1/S - 1)d) est.e) remarks (x) (y) 6-7 0.56 0.102 3 3.38 0.0001 9.07 - not used 7-8 0.67 0.109 143 7.18 0.0081 4.81 0.02 used to estimate St 8-9 0.78 0.116 271 7.76 0.0229 3.75 0.02 9-10 0.90 0.125 318 7.86 0.041 3.15 0.04 10-11 1.03 0.134 416 8.04 0.087 2.58 0.08 11-12 1.17 0.146 488 8.11 0.168 1.60 0.17 12-13 1.32 0.160 614 8.25 0.362 0.67 0.34 13-14 1.49 0.177 613 8.15 0.666 -0.69 0.59 used to estimate Z (see Table 4.4.5.1) 14-15 1.67 0.197 493 7.83 1.020 - 0.81 15-16 1.88 0.223 278 7.13 - - 0.94 16-17 2.12 0.257 93 5.89 - - 0.99 17-18 2.40 0.303 73 5.48 - - 1.00 18-19 2.74 0.370 7 2.94 - - 1.00 19-20 3.15 0.473 2 1.44 - - 1.00 20-21 3.70 0.659 2 1.11 - - 1.00 not used too close to L¥ 21-22 4.53 1.094 0 - - - 1.00 22-23 6.19 4.094 1 -1.40 - - 1.00 23-24 - - 1 - - - 1.00 K = 0.59 per year,L¥ = 23.1 cm,t0 = -0.08 year Selection regression: a = T1 = 8.7111-b = T2 = 6.0829t50% = 8.7111/6.0829 = 1.432t75% = (ln (3) + 8.7111)/6.0829 = 1.613L50% = 23.1 * [1 - exp(0.59 * (-0.08 - 1.432))] = 13.6 cmL75% = 23.1 * [1 - exp(0.59 * (-0.08 - 1.613))] = 14.6 cmSt est. = 1/[1 + exp (8.7111 - 6.0829 * t)]

Exercise 6.7 Using a selection curve to adjust catch samples

L50% = 13.6 cm
S1 = 13.6 * ln (3)/(14.6 - 13.6) = 14.941

L75% = 14.6 cm
S2 = ln (3)/(14.6 - 13.6) = 1.0986
SL = 1/[1 + exp (14.941 - 1.0986 * L)]

Worksheet 6.7

 length groupL1 - L2 mid point observed biased sample selection ogiveSL estimated unbiased sample 6-7 6.5 3 0.00041 7326a) 7-8 7.5 143 0.00123 116491 8-9 8.5 271 0.00367 73769 9-10 9.5 318 0.01094 29067 10-11 10.5 416 0.03212 12952 11-12 11.5 488 0.09054 5390 12-13 12.5 614 0.2300 2670 13-14 13.5 613 0.4726 1297 14-15 14.5 493 0.7288 676 15-16 15.5 278 0.890 312 16-17 16.5 93 0.960 97 17-18 17.5 73 0.986 74 18-19 18.5 7 0.995 7 19-20 19.5 2 0.998 2 20-21 20.5 2 0.999 2 21-22 21.5 0 1.000 0 22-23 22.5 1 1.000 1 23-24 23.5 1 1.000 1 a) 3/0.00041 = 7326

Fig. 18.6.7 Biased sample of goatfish and estimated unbiased sample, corrected for selectivity. Data source: Ziegler, 1979. (see Worksheet 6.7)

Exercise 7.2 Stratified random sampling versus simple random sampling and proportional sampling

Worksheet 7.2

 stratumj s (j) s (j)2 N (j) 1 large 28.906 835.57 10 25413 423 2 medium 8.569 73.43 30 9091 457 3 small 2.809 7.89 60 1524 252 total 100 36028 1132 a) Simple random sampling b) Proportional sampling

c) Optimum stratified sampling

 stratumj s(j) * N(j) 1 large 289.06 0.40 8 2 medium 257.07 0.36 7 3 small 168.55 0.24 5 Total 714.68 1.00 n = 20 Comparison of results random proportional optimum 3.06 2.10 1.20 allocation per stratum 1 large ? 2 8 2 medium ? 6 7 3 small ? 12 5

Exercise 8.3 The yield per recruit model of Beverton and Holt (yield per recruit, biomass per recruit as a function of F)

Worksheet 8.3

 Tc = Tr = 0.2 Tc = 0.3 Tc = 1.0 F Y/R B/R Y/R B/R Y/R B/R 0.0 0.00 8.28 0.00 8.00 0.00 4.53 0.2 1.36 6.81 1.33 6.67 0.79 3.96 0.4 2.28 5.71 2.26 5.65 1.41 3.51 0.6 2.91 4.85 2.92 4.86 1.89 3.15 0.8 3.34 4.18 3.39 4.24 2.28 2.85 1.0 3.64 3.64 3.73 3.73 2.60 2.60 1.2 3.84 3.20 3.98 3.31 2.86 2.39 1.4 3.97 2.84 4.15 2.97 3.08 2.20 1.6 4.06 2.54 4.28 2.68 3.27 2.05 1.8 4.11 2.28 4.38 2.43 3.43 1.91 2.0 4.14 2.07 4.44 2.22 3.57 1.79 2.2 4.15 * 1.88 4.49 2.04 3.69 1.68 2.4 4.14 1..73 4.51 1.88 3.80 1.58 2.6 4.13 1.59 4.53 1.74 3.89 1.50 2.8 4.10 1.47 4.54 1.62 3.98 1.42 3.0 4.08 1.36 4.54 * 1.51 4.05 1.35 3.5 4.00 1.14 4.52 1.29 4.21 1.20 4.0 3.91 0.98 4.48 1.12 4.33 1.08 4.5 3.82 0.85 4.44 0.99 4.42 0.98 5.0 3.74 0.75 4.39 0.88 4.50 0.90 100.0 2.39 0.02 3.35 0.03 5.15 * 0.05 *) MSY/R

MSY increases when Tc increases, because more fish survive to a large size before they are caught. From age 0.2 years to age 1.0 years the biomass production caused by individual growth exceeds the loss caused by the death process. This, of course, is not true for any high value of Tc. If, for example, Tc would be larger than the lifespan of the species in question, no fish would be caught.

curve A: (Tc = 0.2) MSY/R = 4.15 (indicated by "*" in the Table)
curve B: (Tc = 0.3) MSY/R = 4.54
curve C: (Tc = 1.0) MSY/R = 5.15

For F = 1 the Y/R is 3.64 (curve A), 3.73 (curve B) or 2.60 (curve C).

Thus, irrespective of the actual mesh size in use an increased yield is expected for an increase of effort (F).

The smaller the actual mesh size the smaller the gain in yield from an effort increment.

Exercise 8.4 Beverton and Holt's relative yield per recruit concept

Worksheet 8.4

 Lc = 118 cm Lc = 150 cm E (Y/R)' (Y/R)' (F) 0 0 0 0 0.1 0.019 0.022 0.020 0.2 0.035 0.043 0.045 0.3 0.048 0.062 0.077 0.4 0.059 0.079 0.120 0.5 0.067 0.093 0.180 = M 0.6 0.071 0.105 0.270 0.7 0.071 *) 0.112 0.42 0.8 0.068 0.116 0.72 0.9 0.063 0.117 *) 1.62 1.0 0.056 0.114 ¥ *) relative MSY/R

Fig. 18.8.3 Yield per recruit and biomass per recruit curves as a function of F at different ages of first capture of ponyfish. Data source: Pauly, 1980

Fig. 18.8.4 Relative yield per recruit curves a a function of exploitation rate (E) for two different values of 50% retention length of swordfish. Data source: Berkeley and Houde, 1980

Exercise 8.6 A predictive age-based model (Thompson and Bell analysis)

Worksheet 8.6

a. No change in fishing effort:

 age group mean weight (g) beach seine mortality gill net mortality natural mortality total mortality stock number beach seine catch gill net catch beach seine yield gill net yield total yield t FB FG M Z '000 CB CG YB YG YB + YG 0 8 0.05 0.00 2.00 2.05 1000 21.3 0 170 0 170 1 283 0.40 0.00 0.80 1.20 129 30.0 0 8486 0 8486 2 1155 0.10 0.19 0.30 0.59 39 2.9 5.7 3383 6428 9810 3 2406 0.01 0.59 0.20 0.80 21 0.15 8.7 356 21002 21358 4 3764 0.00 0.33 0.20 0.53 9.7 0 2.5 0 9312 9312 5 5046 0.00 0.09 0.20 0.29 5.7 0 0.44 0 2241 2241 6 6164 0.00 0.02 0.20 0.22 4.3 0 0.08 0 471 471 7 7090 0.00 0.00 0.20 0.20 3.4 0 0 0 0 0 total 54.35 17.42 12395 39454 51848

b. Closure of the beach seine fishery:

 age group mean weight (g) beach seine mortality gill net mortality natural mortality total mortality stock number beach seine catch gill net catch beach seine yield gill net yield total yield t FB FG M Z '000 CB CG YB YG YB + YG 0 8 0.00 0.00 2.00 2.00 1000 0 0 0 0 0 1 283 0.00 0.00 0.80 0.80 135 0 0 0 0 0 2 1155 0.00 0.19 0.30 0.49 61 0 6.9 0 10550 10550 3 2406 0.00 0.59 0.20 0.79 39 0 16.0 0 36560 36560 4 3764 0.00 0.33 0.20 0.53 17.8 0 4.6 0 16301 16301 5 5046 0.00 0.09 0.20 0.29 10.5 0 0.8 0 3923 3923 6 6164 0.00 0.02 0.20 0.22 7.8 0 0.14 0 824 824 7 7090 0.00 0.00 0.20 0.20 6.3 0 0 0 0 0 total 0 28.44 0 68158 68158

Although total yield increased in the case of closure of the beach seine fishery, a closure of this fishery without considering the socio-economic aspects is not recommended.

Exercise 8.7 A predictive length-based model (Thompson and Bell analysis)

Worksheet 8.7

 length class mean biomass catch yield value L1 - L2 F(L1, L2) N(L1) * D t C(L1, L2) (L1, L2) (L1, L2) 10-15 0.03 1000 6.47 9.94 0.19 0.19 15-20 0.20 890.56 17.02 63.54 3.40 3.40 20-25 0.40 731.70 31.97 112.28 12.79 19.18 25-30 0.70 535.20 45.18 152.08 31.62 47.44 30-35 0.70 317.95 50.39 102.75 35.27 70.55 35-40 0.70 171.15 48.27 64.08 33.79 67.59 40 - L¥ 0.70 79.60 61.10 55.72 42.77 85.55 Totals 260.44 560.39 159.86 293.91

Exercise 8.7a A predictive length-based model (Yield curve, Thompson and Bell analysis)

Worksheet 8.7a

 length class mean biomass catch yield value L1 - L2 F(L1, L2) N(L1) * D t C(L1, L2) (L1, L2) (L1, L2) 10-15 0.06 1000 6.44 19.79 0.38 0.38 15-20 0.40 881.22 16.25 121.30 6.50 6.50 20-25 0.80 668.94 27.08 190.22 21.66 32.50 25-30 1.40 407.39 29.97 201.75 41.95 62.93 30-35 1.40 162.40 22.02 89.80 30.82 61.65 35-40 1.40 53.36 12.56 33.36 17.59 35.19 40 - L¥ 1.40 12.84 5.80 10.57 8.12 16.24 Totals 120.13 666.79 127.05 215.41

Fig. 18.8.7A Thompson and Bell analysis, prediction of mean biomass, yield and value (values for X = 1 and X = 2 correspond to those calculated on Worksheets 8.7 and 8.7a respectively)

Exercise 9.1 The Schaefer model and the Fox model *)

Worksheet 9.1

 year yield (tonnes) headless effort Schaefer Fox i Y(i) f(i)(x) Y/f(y) ln (Y/f)(y) 1969 546.7 1224 447 6.103 1970 812.4 2202 369 5.911 1971 2493.3 6684 373 5.922 1972 4358.6 12418 351 5.861 1973 6891.5 16019 430 6.064 1974 6532.0 21552 303 5.714 1975 4737.1 24570 193 5.263 1976 5567.4 29441 189 5.242 1977 5687.7 28575 199 5.293 1978 5984.0 30172 198 5.288 mean value 17286 305.2 5.666 standard deviation 11233 102.9 0.3558 intercept (Schaefer: a, Fox: c) 444.6 6.1508 slope (Schaefer: b, Fox: d) -0.008065 -0.000028043 variance of slope:sb2 = [(sy/sx)2 - b2]/(10 - 2) 2.361 * 10-6 2.7113 * 10-11 standard deviation of slope, sb 0.0015364 0.000005207 Student's distribution t10-2 2.31 2.31 confidence limits of slope: b + tn-2 * sb upper -0.0045 -0.00001601 b - tn-2 * sb lower -0.0116 -0.00004007 variance of intercept: 973.4 0.01152 standard deviation of intercept 31.20 0.1073 confidence limits of intercept: a + tn-2 * sa upper 517 6.40 a - tn-2 * sa lower 372 5.90 MSY Schaefer: -a2/(4b) 6128 tonnes MSY Fox: -(1/d) * exp (c - 1) 6154 tonnes fMSY Schaefer: -a/(2b) 27565 boat days fMSY FOX: -1/d 35660 boat days *) a, b replaced by c, d for the Fox-model

Worksheet 9.1a

 fboat days Schaeferyield (tonnes) Foxyield (tonnes) 5000 2021 2039 10000 3640 3544 15000 4854 4620 20000 5666 5354 25000 6074 5817 fMSY 6128 = MSY 30000 6080 6068 35000 5681 6153 fMSY 6154 = MSY 40000 4880 6112 45000 3675 5976

Exercise 13.8 The swept area method, precision of the estimate of biomass, estimation of MSY and optimal allocation of hauls

Worksheet 13.8

STRATUM 1:

 CPUE VESSEL TRAWL CURRENT DIST AREA CPUA haul no. Cw/tkg/h speedknots coursedeg. w. spr.m speedknots dir. deg. nm. sweptsq.nm. Cw/a = Cakg/sq.nm. i VS dir V h * X2 CS dir C D a 1 7.0 2.8 220 18 0.5 90 2.508 .02438 287.2 2 7.0 3.0 210 16 0.5 180 3.442 .02974 235.4 3 5.0 3.0 200 17 0.3 135 3.139 .02881 173.6 4 4.0 3.0 180 18 0.4 230 3.271 .03180 125.8 5 1.0 3.0 90 17 0.5 270 2.500 .02295 43.6 6 4.0 3.0 45 18 0.4 160 2.854 .02774 144.2 7 9.0 3.5 25 18 0.4 200 3.102 .03015 298.5 8 0.0 3.0 210 18 0.3 300 3.015 .02930 0.0 9 0.0 3.5 0 18 0.4 0 3.900 .03790 0.0 10 14.0 2.8 45 18 0.6 0 3.252 .03161 442.9 11 8.0 3.0 120 18 0.3 300 2.700 .02624 304.9

STRATUM 2:

 CPUE VESSEL TRAWL CURRENT DIST AREA CPUA haul no. Cw/tkg/h speedknots coursedeg. w. spr.m speedknots dir. deg. nm. sweptsq.nm. Cw/a = Cakg/sq.nm. i VS dir V h * X2 CS dir C D a 12 42.0 4.0 30 17 0.5 160 3.698 .03395 1237.1 13 98.0 3.3 215 17 0.4 90 3.088 .02835 3457.3 14 223.0 3.9 30 17 0.0 0 3.900 .03580 6229.2 15 59.0 3.8 35 17 0.3 180 3.558 .03266 1806.3 16 32.0 3.5 210 17 0.5 270 3.775 .03465 923.5 17 6.0 2.8 210 17 0.5 330 2.587 .02374 252.7 18 66.0 3.8 45 17 0.5 30 4.285 .03933 1678.0 19 60.0 4.0 30 18 0.5 180 3.576 .03475 1726.5 20 48.0 4.0 210 18 0.5 180 4.440 .04315 1112.3 21 52.0 3.8 20 18 0.4 180 3.427 .03331 1561.3 22 48.0 4.0 30 18 0.5 190 3.534 .03435 1397.4 23 18.0 3.0 210 18 0.3 190 3.284 .03192 563.9

 confidence limits of : stratum number of haulsn s s/Ö n Student's distr.t (n - 1) confidence limits for 1 11 186.9 141.6 42.7 2.23 [92, 282] 2 12 1828.8 1597.5 461.2 2.20 [814, 2843]

Mean biomass for total area:
Area of stratum 1 and 2 combined: A = A1 + A2 = 24 + 53 = 77 sq.nm.

Total biomass of whole area: B(A) = 1317.0 * 77/0.5 = 202818 kg, say 203 tons

From Eq. 9.3.1: MSY = 0.5 * 0.6 * 203 = 61 tons/year.

Worksheet 13.8a (for plotting graph maximum relative error)

 number of hauls tn-1 stratum 1 stratum 2 n e a) e b) 5 2.78 0.94 1.09 10 2.26 0.54 0.62 20 2.09 0.36 0.41 50 2.01 0.22 0.25 100 1.98 0.15 0.17 200 1.97 0.11 0.12

Worksheet 13.8b (optimum allocation)

 stratum s A A * s A * s/S A * s 200*A*s\S A*s 1 141.6 24 3398 0.039 8 hauls 2 1597.5 53 84670 0.961 192 hauls Total 88068 200 hauls

In Part 1: Manual, a selection of methods for fish stock assessment are described in detail, with examples of calculations. Special emphasis is placed on methods based on the analysis of length frequencies. After a short introduction to statistics, the manual covers the estimation of growth parameters and mortality rates; virtual population methods, including age-based and length-based cohort analysis; gear selectivity; sampling; prediction models, including Beverton and Holt's yield-per-recruit model and Thompson and Bell's model; surplus production models; multispecies and multifleet problems; the assessment of migratory stocks; plus a discussion on stock/recruitment relationships and demersal trawl surveys, including the swept-area method. The manual ends with a review of stock assessment, giving an indication of methods to be applied at different levels of availability of input data, a review of relevant computer programs produced by or in cooperation with FAO, and a list of references. In Part 2: Exercises, a number of exercises are given with solutions. These exercises are directly related to the various chapters and sections of the manual.